Q UA N T U M N O I S E I N T H E S P I N T R A N S F E R T O R Q U E E F F E C T PhD student: Camillo Tassi Supervisor: prof. Roberto Raimondi Coordinator: prof. Giuseppe Degrassi Dottorato di Ricerca in Fisica – XXXI Ciclo Dipartimento di Matematica e Fi

Q U A N T U M N O I S E I N T H E S P I N T R A N S F E R T O R Q U E E F F E C T PhD student: Camillo Tassi

Supervisor: prof. Roberto Raimondi Coordinator: prof. Giuseppe Degrassi

Dottorato di Ricerca in Fisica – XXXI Ciclo Dipartimento di Matematica e Fisica

Università degli Studi Roma Tre

It appeared to me that there were two paths to truth

— Georges Lemaître

A C K N O W L E D G M E N T S

I want to give a special thanks to the supervisor, Roberto Raimondi (Dipartimento di Matematica e Fisica – Roma Tre), and to Marco Barbieri (Dipartimento di Scienze – Roma Tre). The thesis work has been produced thanks to their patience and crucial contributions.

Due to the deadline, I could not do a further thesis revision and I am very grateful to the referees for any suggestions they will make for the final review.

I also want to thank all the people I worked with and talked about physics during these years: the people who gave me the chance to work on the plasma physics; in particular, Brunello Tirozzi, Renato Spigler, Paolo Buratti, Alessandro Cardinali and Giovanni Montani.

Those who gave me the chance to work and discuss on cellular au- tomata; in particular (besides Marco Barbieri), Alessio Serafini, Marco Genoni, Federico Centrone and Michele Avalle. The people who gave me the chance to work at the “Dipartimento di Architettura” and those I worked with; in particular, Laura Tedeschini Lalli, Valerio Ta- lamanca, Fabio Bruni and Giuditta Bravaccino. The people who gave me the chance to work at the “LS-OSA” project; in particular (besides Roberto Raimondi), Settimio Mobilio, Ilaria De Angelis and Marco Valli. Furthermore I want to thank the people of the “laboratorio di liquidi”, my colleagues and, in particular, the “NEQO” group.

Finally a special thanks to my family and friends that have always supported me and, last but not least, to Denise.

iii

C O N T E N T S

1 i n t r o d u c t i o n 1

2 p r e c e s s i o n p h e n o m e n o l o g i c a l t h e o r i e s 9 2.1 Landau–Lifshitz–Gilbert equation 9

2.2 Thermal fluctuactions 12

3 m i c r o s c o p i c d e r i vat i o n s o f t h e d y na m i c s 17 3.1 The Slonczewski-Berger term 17

3.2 Open quantum systems 19

3.3 Microscopic models for the magnet dynamics 23 4 n o n-equilibrium formalism 29

4.1 Generalities on many-body physics 29

4.1.1 Creation and annihilation operators 29 4.1.2 Observables 31

4.2 Keldysh formalism 33

4.2.1 Keldysh contour motivation 33 4.2.2 Coherent states 35

4.2.3 Keldysh path integral 39 4.2.4 Two-point Green function 40 4.2.5 Keldysh rotation 45

4.2.6 Expansions and Wick theorem 48 5 t h e m a n y-body model 51

5.1 Holstein-Primakoff bosonization 52 5.2 The many-body Hamiltonian 53 5.3 Initial state 55

5.4 H_bHamiltonian component 56 6 t h e c l a s s i c a l c u r r e n t 59 7 k e l d y s h a c t i o n 63

7.1 Magnet-electrons action 63 7.2 Magnet action 65

8 l i n e a r t e r m s i n b 69 8.1 Action 69

8.2 Classical limit 69 8.3 Equation of motion 71

9 q ua d r at i c c o r r e c t i o n s i n b 73 9.1 Corrections to the motion equation 73 9.2 One fermionic propagator 76

9.3 cl-q two fermionic propagator 77 9.4 q-q two fermionic propagator 81

9.5 Comparison with Brown thermal noise 84 9.6 Numerical example 84

9.6.1 Quantum corrections 87 10 p o s s i b l e m o d e l e x t e n s i o n s 89

v

Appendix

a g i l b e r t d a m p i n g t e r m 93 a.1 Action S¹ 93

a.2 S¹corrections in the dynamics equation 95 b i b l i o g r a p h y 97

1

I N T R O D U C T I O N

The central topic of the thesis work is the quantum noise in the spin transfer torque, which is a spintronics effect.¹In particular in figure1.1 on the following pagewe can see some of the most studied spintronics applications: the giant magnetoresistance (GMR), in which we have a current in a ferromagnetic layer. Here the electrons with spin parallel to the ferromagnet magnetizazion feel a lower resistance with respect to the anti-parallel spin electrons. In the spin transfer torque effect (STT), a polarized current produces a torque in the magnetization of a ferromagnet. In the spin Hall effect (SHE), we see a spin accumulation on the lateral surfaces of an electric paramagnetic conductor, due to the spin-orbit interaction. Finally, in a paramagnetic conductor, we can see in the figure a current-induced spin polarization (CISP) [41].

In particular the GMR is the effect that allows us to read the bits in the hard drives (each bit is encoded in a ferromagnetic layer), while the STT allows to write a hard disk bit.

In a typical STT device a hard ferromagnet (its magnetizazion does not rotate) polarizes a current that exerts a torque on the magneti- zazion of a second ferromagnet [51] (see figure1.2 on the next page).

In recent years, the advances in fast time-resolved measurements have showed that the magnetization dynamics of a nanomagnet crossed by a polarized current presents a stochastic behaviour at short time interval [11,13,14,59].

If we want fast and small devices the stochastic behaviour becomes important. We need to study it because:

• we do not want that the noise disturbs the device working,

• we can engineer the noise to help the magnet switching, without increasing the current (to increase the current means to increase dissipative effects and then heat up the device) [32].

One of the first heuristic equation considered for the magnetization dynamics, is the Landau-Lifshitz-Gilbert equation [21] that describes the magnetization dynamics in presence of an external magnetic field H:~

∂_M~

∂t =γ˜ _M~ × ~^H−^λ_M~ × ( ~^M× ~^H). (1.1) If λ = 0 we simply have the Euler equation (see the figure 2.1a on page 10); the term proportional to λ is perpendicular to both _M~

1 “Spintronics” is a contraction between “spin” and “electronics”. In particular it studies the electronic devices in which the information is carried by both the charge and the spin of electrons.

1

GMR

STT CISP

SHE

Figure 1.1: In this figure, taken from the reference [41], some of the most studied spintronics effects are condensed.

Ferro 2 Ferro 1

Figure 1.2: The scheme of a typical spin transfer torque device: an unpolar- ized current comes from left; the first ferromagnet polarizes the current. Then the polarized current exerts a torque on the second ferromagnet.

i n t r o d u c t i o n 3

¯x spin-up electron

spin-down electron

~p

~p

~s

~p ~p

~s

∆µs

~s

~s

~s

spin potential difference s =±1

~p

nano-magnet

~J

Figure 1.3: We considered a spin potential difference that induces a spin current, flowing trough the magnet.

and the precession direction and then describes the damping (the magnetization tends to align to the external magnetic field – see figure 2.1b on page 10). Finally, to take into account the thermal fluctuactions, Brown [7] introduced a stochastic magnetic field~_h(t)in the equation:

∂_M~

∂t =γ˜ _M~ × (~^H+~h(t))−^λ_M~ × [ ~^M× (~^H+~h(t))]

(see figure2.1c on page 10).

Our aim in the thesis work is

• to find a microscopically derived equation that could describe the magnetization dynamics in presence of a polarized current,

• to describe the thermal and the quantum noise induced in the magnet motion by the electric current;

• to employ a formalism that is easily generalizable (electron- electron interaction, presence and interaction of several magnets, etc.).

To obtain that, we considered the setup described in figure1.3and we employed the Keldysh formalism.

The interaction model between the magnet and the electrons that we have considered is very simple [63]:

H= − ^∂

2¯x

2 m+γ~_B·~^J+δ(¯x) (λ₀+λ~_J·~^s),

where the Hamiltonian first term is the electron kinetic energy,~_{B is} a weak external magnetic field,~s is the electron spin,~J is a spin that represents the magnet degrees of freedom and the delta function fixes the magnet position at ¯x =0. Figure1.4 on the following pageshows the potential seen by the electron.

To treat the problem with the many-body Keldysh formalism, we considered the magnet degrees of freedom as a bosonic system, thanks

transmitted electron

¯x = 0 ¯x

incident electron

potential seen by an electron with spin parallel to the nano-magnet spin

reflected electron

potential seen by an electron with spin anti-parallel to the nano-magnet spin p

V+

V₋

Figure 1.4: An electron that comes from left to right feels a potential V+:= d· (^λ0+J λ/2)and V₋ :=d· (^λ0−^{J λ/2}), depending whether its spin is parallel or anti-parallel to the magnetization, where d is the magnet length.

(a) Two-boson vertex. (b) One-boson vertex.

Figure 1.5: Feynman vertices.

to the Holstein-Primakoff bosonization (that is explained in detail in the section5.1 on page 52). With this representation, the interaction between the magnet and the electrons can be depicted by the Feynman vertices in figure1.5: the one-boson vertex is of the order 1/√

J, while the two-boson vertex is of the order 1/J (this is considered in the section5.4 on page 56). For a typical nanomagnet J∼^{104 [63], so we} will consider only the terms up to the 1/J-order, that is, Feynman diagrams with a single one-boson vertex (order of magnitude 1/√

J) and Feynman diagrams with a couple of one-boson vertices or with a single two-boson vertex (order of magnitude 1/J); see figures1.6.

(a) (b)

(c)

Figure 1.6: Feynman diagrams up to the 1/J order. The “tadpole” diagram is 1/√

J-order, while the other two are 1/J-order.

i n t r o d u c t i o n 5

t +

−

U (t)

U (t)^†

G⁺⁻

G⁻⁺

G⁺⁺

G⁻⁻

Figure 1.7: The Keldysh time contour with the propagators.

We then applied the Keldysh formalism, that allowed us to write the observables in term of functional integral:

h^Oˆ(t)i =^{trh ˆU}t,t^†0O ˆˆ Ut,t₀ ˆρ(t₀)ⁱ=

=

Z

D[¯b+, ¯b₋, ¯ψ+, ¯ψ₋, b+, b₋, ψ+, ψ₋]·

·^O(¯b+, ¯b₋, ¯ψ+, ¯ψ₋, b+, b₋, ψ+, ψ₋)e^{i S(¯b+,¯b−, ¯ψ+, ¯ψ−,b+,b−,ψ+,ψ−)}, (1.2) where b (complex numbers) refers to the magnet degrees of freedom and ψ (Grassmann numbers) to the electronic ones; S is the Keldysh action. ˆU_t,t^†₀ =U^ˆt₀,t and then in the Keldysh formalism we have both a forward and backward in time path, with four propagators (see1.7).

In particular the indices±for the paths t→^b±(t)and t→^ψ±(t)refer to the time direction. The Keldysh rotation reduces the propagators number from four (not independent) to three; for the bosons the Keldysh rotation is given by

b^cl = ^b

++b⁻

√2 , b^q= ^b

+−^b−

√2 , (1.3a)

¯b^cl = ^¯b

++¯b⁻

√2 , ¯b^q= ^¯b

+−^¯b−

√2 ; (1.3b)

note that, like in the Feynman path integral case, in the classical limit we have a single path (for both forward and backward time direction) and then b^q=0 (that justifies the name quantum and classical part for b^q and b^cl respectively); anyway the formalism will be discussed in the section4.2 on page 33.

To obtain the magnet equation of motion, we traced over the elec- tronic degrees of freedom, getting terms that can be represented by the Feynman diagrams in figures 1.6 on the preceding page. In this way, we found a functional integral expression for the magnet observables of the form:

h^Oˆ(t)i =^{trh ˆU}t,t^†0O ˆˆ U_t,t0 ˆρ(t₀)ⁱ=

Z

D[I₁, I₂]e^−RdtI21

(t)+I22(t)

2 ·

·

Z

D[¯b^cl, ¯b^q, b^cl, b^q]O(¯b^cl, b^cl)eⁱ{^{Rdt ¯bq}[^{i ∂}tb^cl+f(b^cl,θ,I1,I2)]^+h.c.}_, (1.4)

where θ is the angle between the current polarization axis and the magnetization direction~J; we will discuss in the thesis the explicit form of the function f , that is the sum of different contributions: in particular, in the chapter8on page 69 we will calculate explicitly the contribution corresponding to the “tadpole” diagram in figure 1.6 on page 4, while in the chapter 9 on page 73 we will consider the other diagrams terms. I₁, I₂are two auxiliary time function that allows us to linearize the Keldysh action with respect to b.²By performing the integration of 1.5with respect to the b^q, ¯b^q, as described in the section8.3 on page 71, we get:

h^Oˆ(t)i =

Z

D[I₁, I₂]e⁻

Rdt^I21(t)+₂^I22(t)Z

D[¯b^cl, b^cl]O(¯b^cl, b^cl)·

·^{δhi ∂}tb^cl+ f(b^cl, θ, I₁, I₂)ⁱδ

h−^{i ∂}t¯b^cl+ ¯f(b^cl, θ, I₁, I₂)ⁱ, (1.5) where δ is the Dirac function. This means that i ∂_tb^cl+f(b^cl, θ, I₁, I₂) = 0 and the complex conjugate are the equations of the motion. t 7→

I_i(t)is a generic function that in the functional integral is weighed by the factor e^{R −I21}

(t)+I22(t)

2 dt. This is the same situation of the Martin- Siggia-Rose action: the weight is the multivariate Gaussian distribution probability, with zero mean value and unitary variance:

h^Ii(t)i =^0, h^Ii(t₁)I_j(t₂)i =^δ(t₁−^t2)δ_ij, (1.6) that is, I_i must be considered as Langevin terms in the equation of motion (see for example the discussion in the section2.2 on page 12).

The stochastic terms presence is not surprising, since it is the typical situation of the open quantum systems: as we will summarize in the section3.2 on page 19, when some degrees of freedom are traced over, a stochastic behaviour appears.³

Finally the equation of motion for the magnet, in terms of micro- scopical quantities, is:

∂t~_J =γ~_B×~^J+



<^C1+ −^{cos θ}=^C2I₁+<^C2I₂ sin θ



ˆz⁰×~^J+ +

=^C1

J +^{cos θ}<^C2I₁+=^C2I₂ sin θ J



~_J× (^ˆz0×~^J) (1.7) where

• Ci depend on the scattering matrix and increase with the spin potential difference∆µs. ˆz⁰ is the current polarization axis. ˆz⁰×~^J

2 that is done thanks to the Hubbard–Stratonovich transformation:

e^−2ax2 =

r 1

2 π a Z

dI e^{−I22 a−i x I}, as described in the chapter9on page 73.

3 the same formalism can explain, for example, the decoherence, where the quantum state collapses randomly in an eigen-energy state.

i n t r o d u c t i o n 7

and ~_J× (^ˆz0× ~^J) are respectively a field-like and a damping- like term (compare with Landau-Lifshitz-Gilbert equation 1.1 on page 1), that produce respectively a precession around the polarization current direction ˆz⁰ and an alignment to it;

• C1 is the contribution of tadpole diagram, while C₂ corresponds to the higher order corrections with respect to 1/√

J (and then disappears in the macroscopic limit J → +∞). C2 contributes also at the zero temperature (both quantum and thermal noise).

Comparing our results with the simpler model in [53] we obtained both the field-like and the damping-like terms and a more complex expression for the noise (note that the field-like and damping-like coefficients are not independent).

The thesis is organized as follow: in the first part the fundamental concepts and techniques are introduced. In particular, in the chapter 2 on page 9the earlier phenomenological theories for the ferromagnet magnetization dynamics are presented. In the chapter3 on page 17 some microscopic theory that describes the interaction between ferro- magnetic layers and polarized currents are described. Finally in the chapter4on page 29the concepts of the Keldysh technique that we need in the following are synthetized.

The second part contains the most original results of the thesis work: in the chapter 5 on page 51 it is described the many-body model that we have considered. In chapter6on page 59we study the relations between the spin potential difference and the spin current.

In the chapter 7on page 63 the Keldysh action is calculated. In the chapter 8on page 69the terms that give rise to the classical equation of motion for the magnetization are considered, while in the chapter 9 on page 73the quantum corrections are evaluated; from that we obtain in particular the quantum noise. Finally in the chapter10on page 89 we draw a possible interesting extension of the model, while the appendixA on page 93describes some terms of the Keldysh action that are typically suppressed, but that give rise to physical interesting interpretations.

2

P R E C E S S I O N P H E N O M E N O L O G I C A L T H E O R I E S

In this chapter we consider the earlier phenomenological theories proposed to describe the magnetization precession in a solid. In par- ticular, we first consider the Landau–Lifshitz–Gilbert equation pro- posed in 1955 by Gilbert (see e. g. the reprinted article [21]), which modifies a previous equation proposed by Landau and Lifshitz in 1935[31]. Finally we consider the thermal fluctuactions for the Lan- dau–Lifshitz–Gilbert equation introduced by Brown [7].

2.1 l a n d au–lifshitz–gilbert equation

In a ferromagnetic material the magnetization is mainly due to the spin of the electrons (one can take into account the contribution of the electrons orbital motion by simply adjusting the value of the gyromag- netic ratio). Below the Curie temperature, the material is divided into elementary domains that are magnetized near the saturation. Then we may divide the material into n cells that are large enough to avoid to consider the microscopic fluctuactions but small enough to take into account the domain structures (that is possible, because a Weiss domain is typically composed by 10¹²−¹⁰¹⁵ atoms). To the i-th cell, a local magnetization field _M~

i is associated.

It is possible to assume for the magnetization the equation of motion:

d_M~

i

dt =γ_M~

i× ~^Hi (2.1)

where

γ= −|^e| 2 me

g, g'^{2 (2.2)}

is the gyromagnetic ratio and _H~_i is an effective field acting on the i-th moment:

H~_i =− ^∂U

∂_M~

i

( ~M₁, . . . ,_M~_n)

(the derivative is intended by components; for example ( ~Hx)_i =

−^∂( ~Mx)_iU).¹

The form of the potential U is established experimentally and for a typical ferromagnet contains five terms: the external magnetic field, the demagnetization energy (that is a self-interaction term), the exchange interaction energy (associated with the gradient in the orientation

1 With the definition2.2of γ,~H is measured in Tesla in the SI.

9

M~i

H~e

(a) Precession motion.

M~_i

~H_e

(b) Damping effect.

M~_i

~H_e

(c) Thermal fluctuactions.

Figure 2.1: The motion of the magnetization_M~_iaround the external magnetic field_H~_e.

of the magnetization), the anisotropy energy (the potential energy depends on the magnetization orientation with respect to the crystal axes), the magnetoelastic energy (deformation effects). In [21] the explicit form of each term is described; for example the external field term is:

U= −

∑

i

M~_i· ~^He,

where_H~_eis the external magnetic field. In particular it is easy to check that, in this case, equation2.1 on the preceding pagereduces to

d_M~

i

dt =γ_M~

i× ~^He,

that is a simple precession motion, like in figure 2.1a. It is convenient to consider the continuous limit:

M~_i

∆~r → ~^M(~r), U( ~M₁, . . . ,_M~_n)

∆~r →^U[ ~M(~r)], ~_Hi → ~^H(~r), where~r is the position in the ferromagnet, ∆~r is the infinitesimal volume and U becomes the density energy functional.

2.1 landau–lifshitz–gilbert equation 11

We argue that the equation of motion2.1 on page 9can be written in the Lagrangian form, as proposed by Gilbert:

d dt

δL[ ~^{M, ˙}_M~]

δ_M~˙ = ^δL[ ~^{M, ˙}_M~]

δ_M~ ^, L[ ~^{M, ˙}_M~] =T [ ~^{M, ˙}_M~]−^U[ ~M]; Gilbert did not fix the form of the kinetic energyT, because he says he was not able to find an expression forT that would correspond to the spin of an elementary particle in quantum mechanics that made physical sense [21].²

If, for example, we consider the case of a fixed external magnetic field, we know from experience that the motion of _M~

i is not simply a precession but, after a while, _M~_i will be oriented along _H~_e; that is, we have some dissipative effects that produce a damping (see figure2.1b on the preceding page). Gilbert introduced the damping effect (for the general case _H~_i) by adding a new term to the Euler-Lagrange equation:

d dt

δL[ ~^{M, ˙}_M~]

δ_M~˙ − ^δL[ ~^{M, ˙}_M~]

δ_M~ + ^δR[_M~^˙]

δ_M~˙ =0, R ^:= ^η 2

Z

d~r ˙_M~ ·_M~^˙ in complete analogy with the motion in a viscous fluid (R ^{is the} Rayleigh dissipation functional). Then the Euler-Lagrange equation is

d dt

δT [ ~^{M, ˙}_M~]

δ_M~˙ −^δT [ ~^{M, ˙}_M~]

δ_M~ +^h−~^H+η_M~˙ⁱ=0 (2.3a)

=⇒ ^∂_M~

∂t =γ_M~ ×

"

~H−^η∂_M~

∂t

#

(2.3b) (indeed note that, even without specifying the form ofT, comparing with equation 2.1 on page 9, we simply have to substitute _{H with}~ H~ −^{η ∂}tM) that is the Landau-Lifshitz-Gilbert equation.~

The Landau-Lifshitz-Gilbert can be put in the original Landau- Lifshitz form [31] simply by redefining the γ coefficient: indeed, by considering the cross product between M and the Landau-Lifshitz-~ Gilbert equation and using the relations~a× (~^b×~^c) = ~b(~a·~^c)−~^c(~a·~^b) and ∂_tM~ · ~^M=0, one obtains immediately an expression for _M~ ×^∂t_M~ in terms of _{M and}~ H. Substituting this expression in the Landau-~ Lifshitz-Gilbert equation one gets:

∂_M~

∂t =γ⁰_M~ × ~^H−^λ_M~ × ( ~^M× ~^H), (2.4a) γ⁰ := ^γ

1+γ²η²| ~^M|^{2, λ:}= ^γ

2η

1+γ²η²| ~^M|^{2, (2.4b)} where the γ⁰_M~ × ~H field term is joined by the−^λ_M~ × ( ~^M× ~^H)damp- ing term, which is orthogonal to both ˆM and the field term (and then produce the damping effect of figure2.1b on the preceding page).

2 in the next chapters we will find a Lagrangian for a magnet _M~

i that is directly obtained from a quantum microscopical derivation.

2.2 t h e r m a l f l u c t ua c t i o n s

Now we want to introduce the thermal fluctuactions in the _{M dynam-}~ ics, by following the reference [7]. To do that, a couple of words on the stochastic processes are useful. In particular, the one-dimensional Brownian motion of a grain in a fluid can be modelled by:

˙v=−^{β v}+A(t) (2.5)

where v is the grain velocity. The interaction between the grain and the fluid is given by two terms: the dynamical friction −^{β v and a} stochastic force per unit mass A(t), such that

• h^A(t)i =^0;

• it is assumed that the evolution governed by the deterministic component, ˙v=−β v, is much slower than the evolution given by the stochastic term. This means that there exists a time interval dt for which v is practically constant in time only considering the deterministic evolution, while, in dt, a rapid variation of A occurs. Typically dt is small with respect to the resolution time and it is a good interval to discretize the time. A is due to the fluid molecules collisions,³ which are pratically independent, if the scattering sections between molecules are small enough.

Then we may consider A(t)and A(t+n dt)(n∈N) independent random variables:

h^A(t₁)A(t₂)i =^{µ δ}(t₁−^t2);

• A(t)are Gaussian random variables. There are some good rea- sons to assume that: the central limit theorem, since the big number of collisions in the time dt, and the fact that, under this assumption, v(t= +∞)is Maxwellian distributed [10].

It follows that W_∆t :=

Z _t+∆t

t A(t)dt,

is a Gaussian process (the superposition of independent Gaussian random variables is a Gaussian random variable) with

h^W∆ti =^0, h^W_∆t² i =^µ∆t

and independent increments; that is a Wiener process. As known, the Wiener process does not have differentiable realizations and the meaning of the equation2.5is simply:

dv=−^{β v dt}+W_dt, (2.6)

3 each collision produces a small variation of v; typically there are few collisions between the grain and the molecules every femtosecond.

2.2 thermal fluctuactions 13

where W_∆t =

Z _t+_∆t

t A(t)dt :=

∑

i

(Wt_i+1 −^Wt_i) is the Itô integral.

From equation2.6 on the preceding pagewe get immediately

h^dvi = −^{β v dt, (2.7a)}

h^dv2i =^{µ dt, (2.7b)}

h^dvni =^O(dt²), n>2, (2.7c)

where v is the velocity at t time, that follows immediately from the moments of Gaussian distribution values. The process is clearly Marko- vian, so we can write the Chapman-Kolmogorov equation:

p(v, t+dt) =

Z

dv⁰p(v⁰, t)p_dt(v|^v0)

where p(v, t)is the probability distribution that the grain velocity is v at time t and p_dt(v|^v0)is the conditional probability distribution to find the velocity equal to v at time t+dt, if it was v⁰ at time t. In particular:

p_dt(v|^v0) =h^δ(v−^dv−^v0)i =

=



1+h^dvi_dv^d₀ +¹

2h^dv2i ^d

2

d(v⁰)² +O(h^dv3i)



δ(v−^v0) =

=δ(v−^v0)−^{β v0 dδ}(v−^v0) dv⁰ dt+ ¹

2µd²δ(v−^v0)

d(v⁰)^{2 dt}+O(dt²), where relations 2.7 have been used. By inserting in the Chapman- Kolmogorov equation, dividing by dt and taking the limit dt→^{0, one} gets:

∂ p

∂t =β∂(p v)

∂v + ¹ 2µ∂²p

∂v², that is the Fokker-Planck equation.⁴

More in general, if we have a stochastic equation of the form dXXXt =µµµ(XXXt, t)dt+σσσ(XXXt, t)dWWWt, (2.9) where XXX_t is an N-dimensional column vector of unknown functions, WW

Wt is an M-dimensional column vector of independent standard

4 The generalization to the three-dimensional case is immediate:

∂ p

∂t =β∇v· (^p~v) +¹

2µ∇²vp. (2.8)

Wiener processes, µµµis called drift vector, σσσis an N×M-dimensional matrix and

DDD := ¹ 2σσσ σσσ^t

is called diffusion tensor, we have that equation 2.9 on the previous pageis equivalent to the probability density equation (Fokker-Planck equation):

∂ p(xxx, t)

∂t =−

∑

∂

∂x_i [µ_i(xxx, t)p(xxx, t)] + +

∑

∂²

∂x_i∂x_j  D_ij(xxx, t)p(xxx, t) . (2.10) Brown (nomen omen) [7] introduced the thermal fluctuactions in the Landau-Lifshitz-Gilbert equation by assuming that the interaction between the magnet M and the thermal bath is modelled by adding a~ stochastic field~_h(t):

∂_M~

∂t =γ_M~ ×

"

H~ +~h(t)−^η∂_M~

∂t

# ,

with the same statistical properties of the force A(t)in the equation2.5 on page 12(see figure2.1c on page 10). In particular

h^hi(t)i =^0, h^hi(t₁)h_j(t₂)i =^µijδ(t₁−^t2)

and Brown principally concentrated on the isotropic case µ_ij = µ δ_ij. As we will see in the next chapters, starting from a microscopi- cal derivation, the interaction between the magnet and an electronic current at temperature T produces terms that are analogous to that introduced phenomenologically by Brown in the equation of motion for _M.~

By applying the same steps to get the Brownian Fokker-Planck equation (here the only complication is that M moves on the sphere~ of radius M_s:=| ~^M|), Brown obtained:

∂W

∂t = ¹ sin θ

∂

∂θ

 sin θ



h⁰ ∂U

∂θ −^g0 _{sin θ}^{1 ∂U}

∂φ



W+k⁰ ∂W

∂θ



+ + ¹

sin θ

∂

∂φ



g⁰ ∂U

∂θ +h⁰ 1 sin θ

∂U

∂φ



W+k⁰ 1 sin θ

∂W

∂φ

 , where

h⁰ = ^η

1/γ²+η²M²_s, g⁰ = ^1/γ

Ms(1/γ²+η²M²_s) and

W(θ, φ)dΩ=W(θ, φ) sin θ dθ dφ= p(θ, φ)dθ dφ,

2.2 thermal fluctuactions 15

with p probability distribution to find M pointing in the direction~ (θ, φ)(azimuthal and polar angle of the spherical coordinate system).

We may expect that the variance µ increases with temperature; to find the relation between µ and T, Brown observed that the canonical statistical distribution

W₀ = A₀e^{−U(θ,φ)v/(kBT)}

(v is the volume of the magnet, since in our definition U is the volume density energy) satisfies the equilibrium (∂tW = 0) Fokker-Planck equation and gives:

µ=2 kBT η/v.

Until now we have considered the phenomenological models intro- duced to describe the magnetization in a ferromagnet. In the next chap- ter we will consider the attempts to finds microscopical derivations of the magnet equation of motion in the form of the Landau-Lifshitz- Gilbert and also the attempts to consider other phenomena, like in the Slonczewski-Berger theory that takes into account the interaction between ferromagnets and electric currents.

3

M I C R O S C O P I C D E R I VAT I O N S O F T H E D Y N A M I C S

Now we are going to consider the microscopic theory that describes the interaction between ferromagnetic layers and polarized currents in the section 3.1 and, after the section 3.2 on page 19 devoted to the introduction to the open quantum systems (that is useful for the next chapters) we will consider two microscopic models (introduced in [62] and [53]) that can describe many aspects of the ferromagnet magnetization dynamics.

3.1 t h e s l o n c z e w s k i-berger term

On 1996, Slonczewski [51] and Berger [3] independently considered how a ferromagnetic layer can polarize an electric current and how a polarized electric current can induce a magnetization rotation on a ferromagnetic layer. This is called spin transfer torque effect and has a large number of technical applications, especially in electronic memory devices (in particular, combined with the giant magnetoresistance effect, has been the leading actor in the rapid storage capacity increase of hard disk drivers in recent years).

In particular Slonczewski considered an electrical conductor com- posed by 5 layers: A, B, C that are paramagnetic layers and F1, F2 ferromagnetic layers crossed by a current along ξ (see fig.3.1 on the following page). The characteristic thickness of the layers is of the order of the nanometer, so that the interlayer exchange coupling can be neglected, but the spin relaxation length is large enough to consider the electronic current as ballistic.

The ferromagnetic layers are seen by electrons as classical potentials with two different values depending on the spin electron orientation (parallel or anti-parallel with respect to the ferromagnet magnetiza- tion).

Slonczewski solved the electron scattering problems in the two regions ξ <0 and ξ > 0 separately and then imposed the matching conditions. To explain better, we can for example consider an electron incident from B onto F2, that in ξ =0 has the spin oriented along _M~₁ (due to the interaction with F1). If_M~₂= ~M₂(t)is the F2 magnetization vector, one can consider the moving reference frame ˆx ˆy ˆz with ˆzk ~^M2

and ˆy k ~^M2× ~^M1. Then, if θ is the angle between_M~_2andM~_1(figure3.1 on the next page), the electron spinor in ξ = 0 (the center of the B region) is (cos(θ/2), sin(θ/2))(see figure3.2 on the following page).

In particular, by assuming that the de Broglie wavelength is short compared with the typical variation length of the potential V_±and in

17

V−

V+

−Q²

−K+²

−K−² eF= 0

ξ =0 ξ1 ξ

A F1 B C

M~1

ˆx ˆz θ M~2

ξ2 V−

V+

F2

Figure 3.1: ξ is the direction of motion for electrons that come from left to right. A, B and C are paramagnetic layers, while F1, F2 are ferromagnets. An electron with spin parallel to the ferromagnet magnetization experiences a potential V₋ in the regions F, while an electron with spin anti-parallel experiences a potential V+. Electrons can have energy between−^{Q2and e}F =0, where, for simplicity, a unit system in which ¯h²/(2 m) = 1 (and m is the electron mass) has been chosen. _M~_1,2is the F1,2 magnetization vector. This figure is present in the reference [51].

|ψi

θ

φ ˆz

ˆy

ˆx

Figure 3.2: The Bloch sphere representation gives a simple geometric interpre- tation of the 1/2-spin systems. In particular, the generic spin state

|^ψi =^α|↑zi +^β|↓zi =^cos(θ/2)|↑zi +^{ei φ sin}(θ/2)|↓zi^{(where α} can be chosen real, since two kets that differ for a multiplicative unitary complex number represent the same quantum state) coin- cides with the state|↑z⁰i^{, where ˆz0}is the direction individuated by the polar angles(θ, φ).

3.2 open quantum systems 19

the parabolic band approximation (that is the electron is considered free inside the layers), Slonczewski found the expression of the electron scattering states (and then of the electron flux) thanks to the WKB approximation.

Since the potential depends on the electron spin, after the scattering the electrons change their spin and also the ferromagnets magnetiza- tions must change direction because of the total angular momentum conservation. This crucial consideration allowed Slonczewski to obtain the equation of motion for the two ferromagnets:

d_M~_1,2

dt = I g ˆM_1,2× (^Mˆ1×^Mˆ2)

where I is the electrons current, ˆM= ~M/| ~^M|^and g= g(θ):=



−⁴+(1+P)³(3+M^ˆ₁·^Mˆ2) 4 P^3/2

−¹

, P := ^K+−^K−

K++K₋

(the K_± parameters are described in the figure3.1 on the preceding page). In particular we must conclude that the presence of a polarized current modifies the Landau–Lifshitz–Gilbert equation.

3.2 o p e n q ua n t u m s y s t e m s

Before going on and describe the models introduced in the last years to obtain a microscopic derivation of the Landau-Lifshitz-Gilbert- Slonczewski equation, a couple of words on the open quantum the- ory [6,48] are useful. Indeed we are going to consider the ferromag- netic layer as a quantum system that is not isolated but interacts with an electric current and an open quantum system is, by definition, a quantum system that interacts with an other system, that we can call environment.

The open quantum theory is useful for different reasons:

• as one can easily imagine, to have a completely isolated quantum system is practically impossible (this is, for example, one of the most important issue in the quantum computing implementa- tions - also known as decoherence problem);

• it can answer to some fundamental problems of the quantum physics; for example:

– why can’t we find the macroscopic systems in a superpo- sition state of two distant positions (or, if you prefer, how does the quantum to classical transition work)? Why in most cases the microscopic quantum systems (like, for ex- ample, the electrons in a solid or in a molecule) are found in

an eigenstate of the Hamiltonian and not in a superposition of two eigenstates?¹

– the quantum theory is causal, that is, if we know the state of a system at a given time (and we know the nature of the interactions), in principle we may evaluate the state of the system at any time. But when we perform a measurement, as known, a stochastic behaviour appears. But how is that possible if we assume that the whole system - quantum system plus measurement apparatus - is a bigger quan- tum system (and then is described by a causal dynamic equation)?

To give a look on how the open quantum theory can approach these questions, we can consider a simple example. But before, we need to introduce the formalism that will be useful also in the next chapters.

In particular we know that, if we have a system in a state |^ψii ^with probability p_i, that is a mixed state, the mathematical representation of the state is given by the density matrix

ˆρ=

∑

i

p_i|^ψii h^ψi|^{, (3.1)}

in the sense that everything we can measure can be obtained from it:

h^Oˆi =^{tr ˆρ ˆO ,}

where ˆO is an observable. The fundamental properties of the density matrix cen be easily checked: ˆρ^†= ˆρ,h^ψ|^ˆρ|^ψi ≥0 (positivity), tr[ˆρ] = 1. A generic linear operator ˆρ that satisfies these three properties is a density matrix; indeed, by using the spectral theorem for self-adjoint operators, we can write the spectral representation

ˆρ=

∑

λ,k

λ|^{λ, k}i h^{λ, k}| ^(3.2)

and verify that λ ≥ ^0, ∑λ,kλ= 1. A crucial point is that the expres- sions3.1and3.2can be different: while ˆρ is the same, it can happens, for example, that the states|^ψiiare not mutually orthogonal. From a physical point of view, this means that we may have two mixed states prepared in different ways that are not distinguishable by simply performing a measurement on the system.

If we have a system that interacts with the environment, we need to consider the Hilbert spaceHS⊗ HE; it is easy to check that

ˆρ_S:=tr_E[ˆρ]

(where ˆρ is a density matrix onHS⊗ HE and tr_E indicates the trace over the environment degrees of freedom) is an operator on theHS 1 Another interesting case is that the chiral molecules are always observed in chiral- ity eigenstates, which are superpositions of different energy eigenstates; the open quantum theory can give an interpretation also for this phenomenon [25,42].

3.2 open quantum systems 21

space that satisfies the density matrixes properties and describes the state of the system S, in the sense that, if ˆO_Sis an observable that acts only on the system S, we have

h^OˆSi = h^OˆS⊗ ^ˆIEi =^trS[ˆρ_SOˆ_S]

(with a slight abuse of notation), where ˆI_E indicates the identity oper- ator overHE.

The time evolution for ˆρ_Sis given by ˆρ_S(t) =trE[U^ˆ(t, t0)ˆρ ˆU^†(t, t0)];

in particular, if at time t0 = 0 the system and environment are not correlated, that is the state is of the form ˆρ(t = 0) = ˆρ_S(0)⊗ ^ˆρE(0), and the spectral decomposition of the initial environment state is

ˆρE(0) =_∑ip_i|^Eii h^Ei|, it is easy to check that ˆρ_S(t) =

∑

ij

p_iE_j

Uˆ(t)E_i ˆρ_S(0)^DE_i  

Uˆ^†(t) _Ej^E=

=

∑

ij

Wˆ_ij ˆρ_S(0)W^ˆ_ij^†, (3.3)

where the ˆW_ij := √p_iE_j

Uˆ(t)E_i operators onHSare called Kraus operators (sometimes the name refers directly toE_j

Uˆ(t)E_i).

Sometimes the equation of motion can be put in the form d

dtˆρ_S(t) =L[^ˆρS(t)]:=−ⁱ[H, ˆρ^ˆ _S(t)] +D[^ˆρS(t)],

that is called master equation. This equation is local in time in the sense that ˆρ_S(t+dt)depends only on ˆρ_S(t). HereL^andD ^{are super-} operators (as we will see immediately, the form of D is well defined under reasonable assumptions), that is linear applications that acts in the space of opearotors on HS. In particular, if D = ^{0, we have} the standard Heisenberg evolution equation and the system can be considered isolated (or the only effect of the environment is to give a particular form to ˆH). If we assume that the evolution ˆρ(t0)7→ ^ˆρ(t) ensures the density matrix positivity and it is trace preserving,²the most general form of the master equation is [6]:

d

dtˆρ_S(t) =−ⁱ[H, ˆρ^ˆ _S(t)]+

− ¹ 2

∑

µ

κ_µn ˆL^†_µˆLµ ˆρ_S(t) + ˆρ_S(t)ˆL^†_µ ˆLµ−^{2 ˆL}µ ˆρ_S(t)ˆL^†_µo ,

κ_µ≥0, that is the Lindblad master equation. Here ˆL_µ are generic opera- tors onHS, whose expression depends on the form of the interaction between the system and the environment.

2 note that, if the master equation is exact, the positivity and the trace preserving are given; but this is not necessarily true when the master equation is obtained by approximations.

electrons source

Figure 3.3: The double slit experiment.

Now we can consider the double slit experiment example: as known, if we consider an electron that crosses a screen with two slits, we will observe interference phenomena (figure 3.3). But if we light up the system (or, if you prefer, if we measure which slit the electron crosses) the interference phenomena disappears and the electron chooses a random slit to cross the screen. It is the electron Hamletic doubt (to be a particle or to be a wave?). The state of the whole system (electron plus light) is pure:

ρ =|^si h^s| =

=

 1

√2|^ψ1, L₁i +√¹

2|^ψ2, L₂i

  1

√2h^ψ1, L₁| +√¹

2h^ψ2, L₂|

 , where|^ψiiis the electron state when the electron crosses the i-th slit (i=1, 2), while|^Liiis the light state. By tracing over the light degrees of freedom, we get

ρ_e=tr_Lρ=

= ¹

2[|^ψ1i h^ψ1| + |^ψ2i h^ψ2|] +¹₂[|^ψ1i h^ψ2| h^L1|^L2i +^h.c.]; we can imagine that a single photon cannot resolve well the electron posistion, that ish^l1|^l2i .1. But, if the light is composed by a large number of photons, that is |^Li = ^NNj=1|^li, where j runs over the photons,

h^L1|^L2i =

∏

h^l1|^l2i = h^l1|^l2i^{N N}−−−→^{→∞ 0} and then

ρ_e' ¹

2[|^ψ1i h^ψ1| + |^ψ2i h^ψ2|]^,

3.3 microscopic models for the magnet dynamics 23

that is we have a mixed state |^ψ1,2i with probability 1/2. Clearly this does not mean that the whole system state is not pure, but the reduced density matrix of electrons is not distinguishable from a mixed state if we operate measurements that involve only the electron observables. Then the coherence (namely the fundamental ingredient of the quantum computation) still exits, but somehow is spread in the environment. By using the open theory standard notation, we say that the environment (the photons) monitors the system, producing a decoherence in one of the two robust (with respect to the interaction system plus environment) states|^ψ1,2i^.

Most quantum systems experience one of the two following envi- ronment monitoring (sometimes these are called classical and quantum limit respectively):

• for macroscopic objects, the typical distance-dependent envi- ronment interaction (for example with photons) gives rise to decoherence into spatially localized wave packets (the eigen- states of the position operator are the robust states). This is for example the case of measurements, since typically the measure- ment apparatus is macroscopic (and not isolated) [25];

• many microsopic systems are found in energy eigenstates, even if the system-environment interaction Hamiltonian depends on observables like the position. This happens when the typical difference of the energy eigenstates of the system is greater than the energies available in the environment (the environment is able to monitor only quantities that are constants of motion) [40].

3.3 m i c r o s c o p i c m o d e l s f o r t h e m a g n e t d y na m i c s

In recent years, the advances in fast time-resolved measurements have showed that the magnetization dynamics of a nanomagnet crossed by a polarized current have a stochastic behaviour at short time inter- vals [11,13,14,59]. Motivated by that, in 2012 Wang and Sham [62, 63] proposed a model that could go behind the classical point of view described by the Landau-Lifshitz-Gilbert-Slonczewski equation. Their idea was that, when the magnet is in the mesoscopic range, some quantum effects can appear.

To take into account the quantum behaviour of the ferromagnet, it is not longer modelled as an external classical field (like in the Slon- czewski model), but as a quantum spin~J. Wang and Sham considered then the Hamiltonian for the magnet plus a single electron given by

H= −¹

2∂²_x+δ(x) (λ₀+λ~s·~^J),

where the electron moves along x, K_e:=−¹₂^∂2xis the electron kinetic energy in a unit system in which ¯h= m=1 (m is the electron mass),

~s is the electron spin, the magnet is placed at x= 0 and d· (^λ0∓^λ) correspond to the potentials V_± in the Slonczewski model (d is the magnet thickness).

The ˆH eigenstates can be easily evaluated by observing that ˆH commutes with the total spin; in particular, if

[ˆJ+ˆs]|J^{, µ}i =^µ|J^{, µ}i^,

[ˆJ+ˆs]²|J^{, µ}i = J (J +¹)|J^{, µ}i^, we have

ψ_k(x) =|J^{, µ}i







L_→e^{i k x}+L_←e^{−i k x}, x <0 R_→e^{i k x}+R_←e^{−i k x}, x >0

=

= |J^{, µ}i







L_→h^x|^ki +^L←h^x| −^ki^{, x} <0 R_→h^x|^ki +^R←h^x| −^ki^{, x} >0 where, since we are interested in the incoming electrons from left to right, R_← =0. To solve the ˆH eingevectors problem means to obtain the relations that allow to evaluate the reflected wave coefficient L_← and the transmitted wave coefficient R_→ as function of L_→, namely the scattering matrix ˆS^.

By using the Clebsch-Gordan coefficients it is possible to write

|J^{, µ}iin terms of|^{m, s}ibasis (where s =±refers to the electron spin and m = −^J, −^J+1, . . . , J to the magnet spin) and then obtain the explicit form of the Kraus operators

Kˆk,s;k⁰,s⁰:=k, s  ˆS

k⁰, s⁰ ,

where|^{k, s}iis the incoming wave, while|^k0 =±^{k, s0} = ±iis the out- going wave. In particular Wang and Sham obtained

Kˆk,s;±^k,s =

 ξ±¹₂



+s ζ ˆJ_z, Kˆk,−^s;±^k,s =ζ ˆJ_s

where ξ and ζ are complex number that depends on(λ0, λ, J, k)and, as usual:

ˆJ_±:= ˆJ_x±^{i ˆJ}y. (3.4)

Before the scattering the state of the electron must be not correlated to the magnet state and Wang and Sham assumed as initial state

ˆρ_in =

∑

ss⁰

f_ss0|^{k, s}i^{k, s0}⊗ ^ˆρJ_in = ˆρ^e_in⊗ ^ˆρ_in^J

so that, after the scattering the magnet state is (see equation3.3 on page 21)

ˆρ_out^J =

∑

±^,s,s0,s00

f_ss0Kˆ±^k,s00;k,s ˆρ_in^J Kˆ^†_±k,s⁰⁰;k,s⁰.